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Algorithmic and cohomological aspects of Aut(Fn) and its subgroups

1.
Normal forms for the group of basis-conjugating automorphisms of a free group (with M. Gutierrez), submitted.

2.
Cohomology of the string group (with B. Bogley), in preparation.



SUMMARY:


In [1] we prove that the group Gn of automorphisms of Fn which take every element of a fixed basis to a conjugate of itself has a regular language of normal forms. (In another guise, Gn appears as the group of motions of n circles in the 3-space and is of some importance in modern physics [BL], [K].) This is a part of an ongoing research motivated by the recent results showing that Aut(Fn) is not (bi)automatic (Bridson-Vogtmann [BV]), while its important subgroups, like the braid groups and the mapping class groups of surfaces are automatic (Thurston [E+], Mosher [M]). It is still to be seen if the methods employed in [1] can be extended to provide an answer to the questions of whether Gn is automatic, and whether Aut(Fn) has a regular language of normal forms.

In [2], we prove that the cohomology ring of the above group Gn has a simple presentation, as conjectured by Brownstein and Lee in [BL]: the generators $\alpha_{ij}\in H^1(G_n)$ $(1\le i,j\le n)$ and relators $\alpha_{ij}^2=0=\alpha_{ij}\alpha_{ji}$ and $\alpha_{ki}\alpha_{kj}=\alpha_{ki}\alpha_{ij}-\alpha_{kj}\alpha_{ji}$. The proof uses the description of abelian subgroups of Gn given in [1], the construction of a constructible complex on which Gn acts with abelian cell stabilizers, and the spectral sequence associated with the action. The proof also gives a description of a free basis of every group Hq(Gn); it is in a bijective correspondence with the set of rooted forests on n vertices which have n-q components. A classical enumeration formula of Cayley implies then that the Poincaré polynomial of Gn is (1+nt)n-1.



REFERENCES:


[BV]
M. Bridson and K. Vogtmann, On the geometry of the automorphism froup of a free group, preprint 1994.

[BL]
A. Brownstein and R. Lee, Cohomology of the group of motions of nstrings in the 3-space, Contemp. Math. 150 (1993), 51-61.

[E+]
D. B. A. Epstein et al., Word Processing in Groups, Bartlett and Jones, Boston 1992.

[L]
H. Kauffman, Knots and Physics, World Scientific 1991.

[M]
L. Mosher, Mapping class groups are automatic, Annals of Math. 142 (1995), 303-384.


next up previous
Next: Automorphisms of free G-groups Up: SUMMARY OF RESULTS Previous: The groups IA(Fn) and GLn

2000-03-16